English

Somos-4 and Somos-5 are arithmetic divisibility sequences

Number Theory 2015-05-04 v1

Abstract

We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalised Somos-4 sequence defined by τn+2τn2=ατn+1τn1+βτn2\tau_{n+2}\tau_{n-2}=\alpha\tau_{n+1}\tau_{n-1}+\beta\tau_n^2 and initial values τ1=τ2=τ3=τ4=1\tau_1=\tau_2=\tau_3=\tau_4=1, that the polynomial τn(α,β)\tau_n(\alpha,\beta) is a divisor of τn+k(2n5)(α,β)\tau_{n+k(2n-5)}(\alpha,\beta) for all n,kZn,k\in\mathbb Z and establish a similar result for the generalized Somos-5 sequence.

Cite

@article{arxiv.1505.00194,
  title  = {Somos-4 and Somos-5 are arithmetic divisibility sequences},
  author = {Peter H van der Kamp},
  journal= {arXiv preprint arXiv:1505.00194},
  year   = {2015}
}

Comments

9 pages

R2 v1 2026-06-22T09:26:38.939Z